PSI - Issue 33

Daniele Gaetano et al. / Procedia Structural Integrity 33 (2021) 1042–1054 Author name / Structural Integrity Procedia 00 (2019) 000–000

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It is worth noting that both the microscopically derived bulk and interface nonlinear behaviors at the macroscale, reported in Figs. 6 and 7, are only valid for a given microstructure, so that they must be recomputed in case of different micro-geometries (including different fiber volume fractions) and moduli contrasts between fiber and matrix phases. The derivation of the homogenized response of the given microstructure has been performed by using a very efficient hierarchical homogenization procedure, involving only off-line computations, which are independent of the given macro-scale problem and, then, can be reused for several failure simulations. After completing this step, a multiscale numerical simulation is performed for obtaining the structural response of a GLARE TM specimen subjected to a pure uniaxial tensile test (under controlled macroscopic strains). The given composite configuration is made of a 130 μm thick glass/epoxy layer (having fiber orientation perpendicular to the applied stress) interposed between two aluminum layers with thickness of 500 μm. Due to the homogeneity of the macroscopic stresses associated with the applied tractions, only a little portion of such a laminated is considered for the multiscale analysis. Moreover, in order to reduce the overall computational time, only a quarter of this geometry is actually modeled, by exploiting the existing double symmetry condition. The mesh adopted for the pre-preg layer is a mapped mesh with element size of 3.0 μm, whereas an unstructured mesh with progressively increasing size towards the (bottom and top) external laminate boundaries is used to reduce the number of Degrees of Freedom (DOFs). In order to capture the failure behavior of this GLARE TM laminate due to transverse cracking and induced ply delamination, a diffuse interface model is also used at the macroscopic scale, so that the pre-preg structured mesh is enriched with cohesive elements placed along the vertical inter-element boundaries (for Mode-I dominated transverse cracking), whereas usual cohesive elements are placed at the physical interface between aluminum and pre-preg layers to account for induced delamination nucleation and evolution. The bulk finite elements of the pre-preg layer are made of nonlinear elastic material, whose homogenized moduli tensor takes the form of Eq. (7), with undamaged moduli obtained from the above mentioned linear homogenization step and damage evolution function reported in Fig. 6B. Besides, the cohesive interfaces embedded in the pre-preg layer are equipped with the microscopically informed traction-separation law, already obtained as an outcome of the aforementioned nonlinear interface homogenization procedure and shown in Fig. 7B. Finally, the interface between aluminum and pre-preg layers is equipped with a traction-separation law of phenomenological type, whose mixed mode fracture properties are listed in Table 3. It is worth noting that the present macro-scale analysis is termed as Multiscale Numerical Simulation (MNS), since the previously derived microscopically informed bulk and cohesive constitutive laws are used as material input at the macroscopic scale, so that a one-way coupling across different scales is implicitly considered. The macro-scale response obtained via the MNS is reported in terms of global stress-strain relation (Fig. 8B) and homogenized secant longitudinal modulus of the pre-preg layer as a function of the applied strain (Fig. 8A). The softening behavior of the laminate due to matrix cracking and inter-ply delamination is clearly visible from both curves (red dotted curves of Fig. 8). As a validation step, the results of this multiscale analysis have been compared with the outcomes of a fully microscopic analysis, denoted as Direct Numerical Simulation (DNS). In this analysis, taken as the reference, all the microstructural details are explicitly modeled, and both fiber/matrix debonding and matrix cracking are included in the same way as in the RUC analysis. The related numerical results, also reported in Fig. 8, show that the proposed multiscale strategy, based on a two-scale cohesive finite element methodology, provides an accurate prediction of the overall composite behavior, especially in terms of the homogenized secant longitudinal modulus of the pre-preg layer. Table 3. Mixed-mode fracture properties for aluminum/pre-preg interface. Materials K n = K s (N/mm 3 ) σ nc (MPa) σ sc (MPa) G I c (N/m) G II c (N/m) Aluminum/pre-preg 1.69×10 9 50 100 50 300